A175342 -id:A175342 - cp's OEIS frontend

A342514 Number of integer partitions of n with distinct first quotients.

1, 1, 2, 2, 4, 5, 6, 8, 11, 14, 18, 24, 28, 35, 41, 52, 64, 81, 93, 115, 137, 157, 190, 225, 268, 313, 366, 430, 502, 587, 683, 790, 913, 1055, 1217, 1393, 1605, 1830, 2098, 2384, 2722, 3101, 3524, 4005, 4524, 5137, 5812, 6570, 7434, 8360, 9416, 10602, 11881

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

Also the number of reversed integer partitions of n with distinct first quotients.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The partition (4,3,3,2,1) has first quotients (3/4,1,2/3,1/2) so is counted under a(13), but it has first differences (-1,0,-1,-1) so is not counted under A325325(13).
The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)
                   (31)   (41)   (42)   (52)    (53)    (63)
                   (211)  (221)  (51)   (61)    (62)    (72)
                          (311)  (321)  (322)   (71)    (81)
                                 (411)  (331)   (332)   (432)
                                        (511)   (422)   (441)
                                        (3211)  (431)   (522)
                                                (521)   (531)
                                                (611)   (621)
                                                (3221)  (711)
                                                        (3321)
                                                        (4311)
                                                        (5211)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A325325.
The ordered version is A342529.
The strict case is A342520.
The Heinz numbers of these partitions are A342521.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts partitions with all adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

A325552 Number of compositions of n with distinct differences up to sign.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 12, 23, 38, 61, 78, 135, 194, 315, 454, 699, 982, 1495, 2102, 3085, 4406, 6583, 9048, 13117, 18540, 26399, 36484, 51885, 72498, 100031, 139342, 192621, 267068, 367631, 505954, 687153, 946412, 1283367, 1745974, 2356935, 3207554, 4311591, 5816404

Offset: 0

Gus Wiseman, May 11 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
a(n) has the same parity as n for n > 0, since reversing a composition does not change whether or not it has this property, and the only valid symmetric compositions are (n) and (n/2,n/2), with the latter only existing for even n. - Charlie Neder, Jun 06 2019

			The differences of (1,2,1) are (1,-1), which are different but not up to sign, so (1,2,1) is not counted under a(4).
The a(1) = 1 through a(7) = 23 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)
       (11)  (12)  (13)   (14)   (15)   (16)
             (21)  (22)   (23)   (24)   (25)
                   (31)   (32)   (33)   (34)
                   (112)  (41)   (42)   (43)
                   (211)  (113)  (51)   (52)
                          (122)  (114)  (61)
                          (221)  (132)  (115)
                          (311)  (213)  (124)
                                 (231)  (133)
                                 (312)  (142)
                                 (411)  (214)
                                        (223)
                                        (241)
                                        (322)
                                        (331)
                                        (412)
                                        (421)
                                        (511)
                                        (1132)
                                        (2113)
                                        (2311)
                                        (3112)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A011782, A070211, A175342, A242882, A325325, A325368, A325404, A325545, A325551, A325553, A325555, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[#]]&]],{n,0,15}]

a(26)-a(42) from Alois P. Heinz, Jan 27 2024

A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 2, 5, 5, 9, 5, 15, 9, 19, 16, 28, 16, 44, 21, 55, 38, 73, 34, 109, 46, 130, 73, 170, 66, 251, 78, 287, 137, 364, 119, 522, 135, 590, 236, 759, 190, 1042, 219, 1175, 425, 1460, 306, 2006, 347, 2277, 671, 2780, 471, 3734, 584, 4197, 1087

Offset: 0

Gus Wiseman, Oct 07 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(2) = 1 through a(12) = 15 partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)      (B)     (C)
            (22)       (33)   (52)  (44)    (63)   (55)     (83)    (66)
                       (42)         (62)    (72)   (64)     (92)    (84)
                       (222)        (422)   (333)  (73)     (722)   (93)
                                    (2222)  (522)  (82)     (5222)  (A2)
                                                   (442)            (444)
                                                   (622)            (552)
                                                   (4222)           (633)
                                                   (22222)          (642)
                                                                    (822)
                                                                    (3333)
                                                                    (4422)
                                                                    (6222)
                                                                    (42222)
                                                                    (222222)

The complement to these partitions is counted by A328164.
The GCD of the divisors of n all minus 1 is A258409(n).
The GCD of the prime indices of n all minus 1 is A328167(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]!=GCD@@(#-1)&]],{n,0,30}]

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686

Offset: 0

Gus Wiseman, Mar 28 2021

nonn

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.

			The a(2) = 1 through a(7) = 14 compositions:
  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
         (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
                (3,1)  (3,2)  (3,3)      (3,4)
                       (4,1)  (4,2)      (4,3)
                              (5,1)      (5,2)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]

\\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

A342498 Number of integer partitions of n with strictly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 6, 8, 9, 12, 12, 14, 16, 18, 20, 24, 26, 27, 30, 35, 37, 45, 47, 52, 56, 61, 65, 72, 77, 83, 90, 95, 99, 109, 117, 127, 135, 144, 151, 164, 172, 181, 197, 209, 222, 239, 249, 263, 280, 297, 310, 332, 349, 368, 391, 412, 433, 457, 480, 503

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

Also the number of reversed integer partitions of n with strictly increasing first quotients.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The partition y = (13,7,2,1) has first quotients (7/13,2/7,1/2) so is not counted under a(23). However, the first differences (-6,-5,-1) are strictly increasing, so y is counted under A240027(23).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)   (32)   (33)   (43)   (44)   (54)
                   (31)   (41)   (42)   (52)   (53)   (63)
                   (211)  (311)  (51)   (61)   (62)   (72)
                                 (411)  (322)  (71)   (81)
                                        (511)  (422)  (522)
                                               (521)  (621)
                                               (611)  (711)
                                                      (5211)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A240027.
The ordered version is A342493.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The strict case is A342517.
The Heinz numbers of these partitions are A342524.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with adjacent x < 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342499 Number of integer partitions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 11, 14, 15, 18, 20, 23, 26, 31, 34, 39, 42, 45, 51, 58, 65, 70, 78, 83, 91, 102, 111, 122, 133, 145, 158, 170, 182, 202, 217, 231, 248, 268, 285, 307, 332, 354, 374, 404, 436, 468, 502, 537, 576, 618, 654, 694, 737, 782, 830

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

Also the number of reversed partitions of n with strictly decreasing first quotients.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The partition (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)  (32)   (33)   (43)   (44)   (54)
                   (31)  (41)   (42)   (52)   (53)   (63)
                         (221)  (51)   (61)   (62)   (72)
                                (321)  (331)  (71)   (81)
                                              (332)  (432)
                                              (431)  (441)
                                                     (531)
                                                     (3321)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A320470.
The ordered version is A342494.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The strict case is A342518.
The Heinz numbers of these partitions are listed by A342525.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with adjacent x < 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A325588 Number of necklace compositions of n with equal circular differences up to sign.

Original entry on oeis.org

1, 2, 3, 4, 4, 7, 5, 9, 8, 10, 8, 17, 9, 14, 15, 22, 12, 23, 14, 31, 23, 25, 19, 48, 25, 35, 36, 56, 33, 59, 43, 86, 64, 74, 76, 136, 95, 127, 138, 219, 178, 245, 249, 372, 370, 445, 506, 747, 730, 907, 1069, 1431, 1544, 1927, 2268, 2981, 3332, 4074, 4896, 6320

Offset: 1

Gus Wiseman, May 11 2019

nonn

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

The circular differences of a sequence c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (11111)  (33)      (34)       (35)
                                     (222)     (1111111)  (44)
                                     (1212)               (1232)
                                     (111111)             (1313)
                                                          (2222)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A049988, A175342, A325549, A325556, A325558, A325590.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&SameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,15}]

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
w(n,s)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==s), t=0, m=1); while(R, R=step(R,n,s); m++; t+=sumdiv(n, d, R[d,k]*d*eulerphi(n/d))/m ); t/n)}
a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, w(n,s))} \\ Andrew Howroyd, Aug 24 2019

Terms a(26) and beyond from Andrew Howroyd, Aug 24 2019

A328164 Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 7, 13, 17, 25, 33, 51, 62, 92, 116, 160, 203, 281, 341, 469, 572, 754, 929, 1221, 1466, 1912, 2306, 2937, 3548, 4499, 5353, 6764, 8062, 10006, 11946, 14764, 17455, 21502, 25425, 30949, 36579, 44393, 52132, 63042, 74000, 88709, 104098, 124448

Offset: 0

Gus Wiseman, Oct 07 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(1) = 1 through a(8) = 17 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (61)       (71)
                    (1111)  (221)    (411)     (322)      (332)
                            (311)    (2211)    (331)      (431)
                            (2111)   (3111)    (421)      (521)
                            (11111)  (21111)   (511)      (611)
                                     (111111)  (2221)     (3221)
                                               (3211)     (3311)
                                               (4111)     (4211)
                                               (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The complement to these partitions is counted by A328163.
The GCD of the divisors of n all minus 1 is A258409(n).
The GCD of the prime indices of n all minus 1 is A328167(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]==GCD@@(#-1)&]],{n,0,30}]

A342497 Number of integer partitions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 25, 32, 36, 43, 49, 60, 65, 75, 83, 96, 106, 121, 131, 150, 163, 178, 194, 217, 230, 254, 275, 300, 320, 350, 374, 411, 439, 470, 503, 548, 578, 625, 666, 710, 758, 815, 855, 913, 970, 1029, 1085, 1157, 1212, 1288, 1360

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

Also called log-concave-up partitions.
Also the number of reversed integer partitions of n with weakly increasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The partition y = (6,3,2,1,1) has first quotients (1/2,2/3,1/2,1) so is not counted under a(13). However, the first differences (-3,-1,-1,0) are weakly increasing, so y is counted under A240026(13).
The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (311)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (411)     (421)      (422)
                                     (3111)    (511)      (521)
                                     (21111)   (4111)     (611)
                                     (111111)  (31111)    (2222)
                                               (211111)   (4211)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A240026.
The ordered version is A342492.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The strict case is A342516.
The Heinz numbers of these partitions are A342523.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with all adjacent parts x <= 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342513 Number of integer partitions of n with weakly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 15, 20, 21, 24, 28, 29, 33, 40, 44, 49, 57, 61, 65, 77, 84, 87, 99, 106, 115, 132, 141, 152, 167, 180, 193, 212, 228, 246, 274, 290, 309, 338, 357, 382, 412, 439, 463, 498, 536, 569, 608, 648, 693, 743, 790, 839, 903, 949

Offset: 1

Gus Wiseman, Mar 17 2021

nonn

Also called log-concave-down partitions.
Also the number of reversed integer partitions of n with weakly decreasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The partition (9,7,4,2,1) has first quotients (7/9,4/7,1/2,1/2) so is counted under a(23).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (111111)  (2221)     (431)
                                               (1111111)  (2222)
                                                          (11111111)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The ordered version is A069916.
The version for differences instead of quotients is A320466.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The strict case is A342519.
The Heinz numbers of these partitions are A342526.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with adjacent parts x <= 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

cp's OEIS Frontend

A342514 Number of integer partitions of n with distinct first quotients.

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A325552 Number of compositions of n with distinct differences up to sign.

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A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

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A342532 Number of even-length compositions of n with alternating parts distinct.

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A342498 Number of integer partitions of n with strictly increasing first quotients.

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A342499 Number of integer partitions of n with strictly decreasing first quotients.

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A325588 Number of necklace compositions of n with equal circular differences up to sign.

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A328164 Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.

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